1.1. THE MODEL 5
assuring that all other terms appearing in our Heff,0 and Heff,1 are of higher order
or trivial, including that Ef (q) Ef is constant.
We end this section with some more technical comments concerning our result
and the difficulties encountered in its proof.
In this work we do not assume the potential to become large away from the sub-
manifold. That means we achieve the confinement solely through large potential
gradients, not through high potential barriers. This leads to several additional tech-
nical difficulties, not encountered in other rigorous results on the topic that mostly
consider harmonic constraints. One aspect of this is the fact that the normal Hamil-
tonian Hf (q) has also continuous spectrum. While its eigenfunctions defining the
adiabatic subspaces decay exponentially, the superadiabatic subspaces, which are
relevant for our analysis, are slightly tilted spectral subspaces with small compo-
nents in the continuous spectral subspace.
Let us finally mention two technical lemmas, which may both be of independent
interest. After extending the pull back metric from a tubular neighborhood of
C in A to the whole normal bundle, NC with this metric has curvature increasing
linearly with the distance to C. As a consequence we have to prove weighted elliptic
estimates for a manifold of unbounded curvature (Lemmas 4.5 & 4.6). Moreover,
since we aim at uniform results, we need to introduce energy cutoffs. A result
of possibly wider applicability is that the smoothing by energy cutoffs preserves
polynomial decay (Lemma 4.11).
1.1. The model
Let (A,G) be a Riemannian manifold of dimension d+k (d, k := {1, 2, · · · })
with associated volume measure dτ. Let furthermore C A be a smooth submani-
fold without boundary and of dimension d/codimension k, which is equipped with
the induced metric g = G|C and the associated volume measure dμ. We will call A
the ambient manifold and C the constraint manifold.
On C there is a natural decomposition T A|C = T C × NC of A’s tangent bundle
into the tangent and the normal bundle of C. We assume that there exists a tubular
neighborhood B A of C with globally fixed diameter, that is there is δ 0 such
that normal geodesics γ (i.e. γ(0) C, ˙ γ (0) NC) of length δ do not intersect.
We will call a tubular neighborhood of radius r an r-tube. Furthermore, we assume
that
(1.5) A and B are of bounded geometry
(see the appendix for the definition) and that the embedding
(1.6) C A has globally bounded derivatives of any order,
where boundedness is measured by the metric G. In particular, these assumptions
are satisfied for A =
Rd+k
and a smoothly embedded C that is (a covering of) a
compact manifold or asymptotically flatly embedded, which are the cases arising
mostly in the applications we are interested in (molecular dynamics and quantum
waveguides).
Let ΔA be the Laplace-Beltrami operator on A. We want to study the Schr¨odinger
equation
(1.7) i∂tψ =
−ε2ΔAψ
+ VAψ
ε
, ψ|t=0
L2(A,dτ)
,
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